Confining-potential energy space (${\Sigma }_V$)

For a nonnegative potential $V(x)\to \infty$ as $|x|\to \infty$, the confining-potential energy space isA local Poincare inequality controls the missing $L^2$ term, so this energy norm makes ${\Sigma }_V$ a Hilbert space.

The embedding ${\Sigma }_V\hookrightarrow L^2({\mathbb{R}}^d)$ is compact. The Rellich-Kondrachov compactness theorem gives compactness on each fixed ball, whilemakes the tails uniformly small.

In two dimensions, the quadratic functionalattains its minimum subject to $\Vert u{\Vert }_4^4=M>0$. A bounded minimizing sequence is weakly compact in the harmonic-oscillator energy space and strongly compact in $L^4$: local Rellich-Kondrachov compactness theorem and the weighted tail bound prevent escape to infinity.

The minimum ofsubject to $\Vert u{\Vert }_2=1$ is attained because ${\Sigma }_V$ embeds compactly into $L^2$. Replacing a minimizer by its absolute value does not increase its energy. The Euler-Lagrange equation therefore produces a nonnegative eigenfunction of $-\Delta +V$ for its lowest eigenvalue.